An elliptical Debye cluster is a system of n identical charged particles interacting through a screened Coulomb potential and confined in a two-dimensional anisotropic parabolic well. A model for this system has been developed with three parameters: n, the well anisotropy α2, and the Debye shielding parameter κ. From numerical solutions of the model, it is shown that the breathing mode persists as the anisotropy increases, and that the normalized, squared breathing frequency increases linearly with α2. Elliptical clusters with n=49 and 15 particles were studied experimentally. The anisotropic potential well was created using a rectangular aperture (17.5×30.2 mm) placed on a flat electrode. The well anisotropy was determined by measuring the center-of-mass (c.m.) oscillation frequencies along the major and minor axes of the resulting elliptical clusters using both driven and thermal oscillations. The two methods give results that are in good agreement. For n=49 particles, the square of the ratios of the c.m. frequencies was α2=2.9±0.1. From a measurement of the breathing frequency, the shielding parameter was found to be κ=2.1±0.2. For n=15 particles, α2=2.8±0.1 and κ=1.9±0.2, demonstrating that an analysis using the model gives consistent results.